Interlacing Polynomials and the Veronese Construction for Rational Formal Power Series

TL;DR

This paper explores the interlacing properties of polynomials derived from formal power series and their applications to Ehrhart polynomials, providing new proofs and identities related to lattice polytopes and colored permutations.

Contribution

It introduces a general interlacing result for polynomial sequences associated with the Veronese construction, offering alternative proofs and new identities in combinatorics and geometric enumeration.

Findings

Proves interlacing preservation under certain polynomial transformations.

Provides a new proof of the interlacing property for refined descent generating functions.

Derives a Carlitz identity for refined colored permutations.

Abstract

Fixing a positive integer $r$ and $0 \le k \le r-1$, define $f^{\langle r,k \rangle}$ for every formal power series $f$ as $ f(x) = f^{\langle r,0 \rangle}(x^r)+xf^{\langle r,1 \rangle}(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := \left( (1+x+\cdots+x^{r-1})^{n} h(x) \right)^{\langle r,k \rangle}$ has only nonpositive zeros for any $r \ge \deg h(x) -k$ and any positive integer $n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ge n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $\left( h^{\langle r,r-i \rangle}(x)\right)_{1\le i \le r}$ is interlacing, so is $\left( U^{n}_{r,r-i}\, h(x) \right)_{1\le i \le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations.